DISCRIMINANTS OF SOME PAINLEVÉ POLYNOMIALS

DAVID P. ROBERTS

Abstract. We derive explicit formulas for the discriminants of the Yablonsky-Vorobiev polynomials P_{m}(x), the biHermite polynomials H_{m,n}(x) and the Okamoto polynomials Q_{m,n}(x), as well as some related resultants. In all three cases, the discriminant and related resultants factor into a product of small primes only. Our formula in the biHermite case reduces when n = 1 to the nineteenth-century formula for the discriminant of a Hermite polynomial. Our other two discriminant formulas do not have direct antecedents.

1. Introduction

The Painlevé hierarchy was discovered by Painlevé and Gambier around 1900. It consists of six second-order non-linear ordinary differential equations I-VI, each depending on appropriate parameters. In brief, one can think of Painlevé VI as a non-linear analog of the usual linear hypergeometric equation. The others are limiting degenerations of Painlevé VI, thus analogs of the confluent hypergeometric equation:

III

↙  ↖
I ← II  V ← VI.
↖  ↙

IV

As one moves rightward in this diagram, one adds a parameter at each step, so that Painlevé I involves 0 parameters and Painlevé VI involves 4.

Most of the large and rapidly-expanding literature on the Painlevé equations is connected with mathematical physics. However, we are looking here at number-theoretic phenomena and familiarity with the literature is not necessary to understand this paper. For readers who nonetheless would like some general background, we recommend [IKSY], [Co], and [Um] as places to start. The textbook [IKSY] characterizes the six Painlevé equations as “the most important non-linear ordinary differential equations.” The instructional conference proceedings [Co] supports this characterization by describing many situations in mathematical physics which ultimately reduce to a Painlevé equation. The article [Um] is an easy-to-read introduction aimed at pure mathematicians.

The bulk of this paper concerns Painlevé II and IV:

(1.1)  y′′ = 2y^{3} + xy + a
(1.2)  y′′ = \frac{(y′)^{2}}{2y} + \frac{3y^{3}}{2} + 4xy^{2} + 2(x^{2} − a)y + \frac{b}{y} .

Around 1960, Yablonsky [Ya] and Vorobiev [Vo] found rational solutions to Painleve II. For m an integer, they defined polynomials P_{m} ∈ Z[x] and

f_{m} = \frac{d}{dx} log (\frac{P_{m+1}}{P_{m}} ) .

Then s_{P} f_{m}(s_{P} x) with s_{P} = 2^{−2/3} is a solution to Painlevé II with a = −m − 1.

In the 1980’s, Murata [Mu] found a family of solutions to Painlevé IV, indexed by (m, n) ∈ Z_{≥0}^{2}. He defined polynomials H_{m,n} and

h_{m,n} = \frac{d}{dx} log (\frac{H_{m,n+1}}{H_{m,n}} ) = \frac{H_{m+1,n}H_{m−1,n+1}}{H_{m,n+1}H_{m,n}} .

The function s_{H}h_{m,n}(s_{H}x) with s_{H} = \sqrt{−2} solves Painlevé IV for a = m + 2n + 1 and b = −2m^{2}. We call the polynomials H_{m,n} biHermite polynomials because they depend on the two indices m and n and H_{m,1}(x) = i^{−m}H_{1,m}(ix) both coincide with the classical Hermite polynomial H_{m}(x).

Also in the 1980’s, Murata [Mu] found another family of solutions to Painlevé IV, this one indexed by (m, n) ∈ Z^{2}. This family forms a generalization of two sequences of polynomials found several years earlier by Okamoto. Murata defined polynomials Q_{m,n} ∈ Z[x] and

g_{m,n} = \frac{d}{dx} log (\frac{Q_{m,n+1}}{Q_{m,n}} ) + x = \frac{Q_{m+1,n}Q_{m−1,n+1}}{Q_{m,n+1}Q_{m,n}} .

Then s_{Q}g_{m,n}(s_{Q}x) with s_{Q} = \sqrt{−2/3} solves Painlevé IV for a = m + 2n and b = (−2/9)(3m − 1)^{2}.

It should be noted that notations and normalizations vary somewhat in the literature. We have opted to keep the most standard form of the Painlevé equations (1.1) and (1.2). Our versions of P_{m}, H_{m,n}, and Q_{m,n} are monic and well-behaved at the primes 2 and 3. These good aspects of our normalizations force the presence of the unattractive irrational scale factors s_{P}, s_{H}, and s_{Q}. However, because of the symmetries (3.2), (4.3), (5.3), all rational solutions are in fact in Q(x).

In §2, we review the formalism of discriminants and resultants. Sections §3, §4, and §5 present our discriminant and resultant formulas for P_{m}, H_{m,n}, and Q_{m,n} respectively. As our title suggests, our primary interest is discriminants. However our inductive method of proof requires us to compute related resultants as well. The induction proceeds by expressing a given discriminant or resultant as a product of earlier discriminants, earlier resultants, and small integers. Accordingly, all the final formulas have a common structure: discriminants and resultants are expressed as products of small integers to large powers. In carrying out the induction, we make fundamental use of quadratic contiguity relations among the polynomials in question.

The polynomials we consider here exhibit great regularity beyond what we capture in our discriminant and resultant formulas. Moreover, the polynomials P_{m} and H_{m,n} are limiting cases of polynomials related to solutions to Painlevé III, V, and VI. In §6 we briefly indicate several ways in which our formulas here are fitting into an emerging larger picture.

We have implemented P_{m}, H_{m,n} and Q_{m,n} in a small Mathematica file. This file can be used as an aid in reading this paper. It is available at

http://cda.morris.umn.edu/~roberts.

2. Discriminants and resultants

Let f(x) = a_{0}x^{M} + ⋯ be a degree M polynomial in Z[x] with complex roots α_{1}, …, α_{M}. Then the discriminant of f is defined to be

(2.1)  Disc(f) = a_{0}^{2M−1} ∏_{1≤i<j≤M}(α_{i} − α_{j})^{2} ∈ Z.

Let g(x) = b_{0}x^{N} + ⋯ be a degree N polynomial in Z[x] with complex roots β_{1}, …, β_{N}. Then the resultant of f and g is

Res(f, g) = a_{M}^{N}b_{N}^{M} ∏_{i=1}^{M} ∏_{j=1}^{N}(α_{i} − β_{j}) ∈ Z. (2.2)

The connection between resultants and discriminants is

Disc(f) = (−1)^{M(M−1)/2}Res(f, f′). (2.3)

While discriminants are just a special case of resultants, they are worthy of special attention.

It is clear from (2.2) that resultants behave homogeneously on constant polynomials, generalize evaluation, are multiplicative, and are symmetric up to sign:

(2.4)  Res(f, c) = c^{deg(f)}
(2.5)  Res(f, x − c) = (−1)^{deg(f)}f(c)
(2.6)  Res(f, g_{1}g_{2}) = Res(f, g_{1})Res(f, g_{2})
(2.7)  Res(f, g) = (−1)^{deg(f)deg(g)}Res(g, f).

Another basic fact about resultants, not so immediate from (2.2), is that

Res(f, qf + r) = Res(f, r). (2.8)

This fact allows one to compute resultants by the Euclidean algorithm. The initial step in all three of our proofs is several applications of (2.8). In these applications qf represents one or two terms in a quadratic relation. So these terms are not seen in our discriminant and resultant calculations.

The relations (2.3)-(2.8) completely characterize the functionals Disc(·) and Res(·, ·). We will not make further reference to (2.1) and (2.2) until §6. Thus we will be working purely over Z, without any reference to complex roots.

Denote the absolute version of discriminants and resultants as follows

D(f) = |Disc(f)|
R(f, g) = |Res(f, g)|.

In the absolute setting, the formalism of discriminants and resultants simplifies as the signs in (2.3), (2.5) and (2.7) disappear. To make our theorems and proofs more readable we will present them in the absolute setting; thus we will ignore minus signs and work with positive numbers only. Of course, it is easier to do the reverse, that is to ignore magnitudes and work with signs only. Doing this yields

Disc(P_{m}) < 0 ⇔ m ≡ 2 (4)
Disc(H_{m,n}) < 0 ⇔ (m, n) ≡ (1, 2), (1, 3), (3, 2), (3, 3) (4)
Disc(Q_{m,n}) < 0 ⇔ (m, n) ≡ \begin{matrix}(2, 0), (2, 1), (2, 2), (2, 3), \\ (1, 2), (1, 3), (3, 0), (3, 1) \end{matrix} (4).

The signs of the resultants we treat are also determined by their indices modulo 4.

3. Yablonsky-Vorobiev polynomials

For m ∈ Z, the Yablonsky-Vorobiev polynomials are defined by

P_{0} = 1
P_{1} = x
(3.1)  P_{m+1}P_{m−1} = P′_{m}P′_{m} − P′′_{m}P_{m} + xP_{m}P_{m}.

From this definition it is apparent that the P_{m} are rational functions. It is not apparent but it is true that they are polynomials. The analogous remarks hold also in the biHermite and Okamoto cases.

The polynomial P_{m} has degree

Δ_{m} = \frac{m(m + 1)}{2}

and satisfies the symmetry

P_{m}(ωx) = ω^{Δ_{m}}P_{m}(x) (3.2)

for ω ∈ C a third root of unity.

Besides the quadratic relation (3.1), we need the following quadratic relations.

(3.3)  P_{m+2}P_{m−2} = (4 − (2m + 1)^{2})P′_{m}P′_{m} +
(x^{4} + (2m + 1)^{2}x)P_{m}P_{m} −
4x^{2}P_{m}P′_{m} + (2m + 1)^{2}P_{m}P′′_{m}
(3.4)  P_{m+2}P_{m−1} = −(2m + 3)P′_{m}P_{m+1} +
(2m + 1)P′_{m+1}P_{m} + x^{2}P_{m}P_{m+1}.

These can be derived from the determinant representation [KYO, Prop. 4.1] of P_{m}, following the model provided by the proof of [KYO, Prop. 2.2].

One has the symmetry P_{m} = P_{1−m} so we will henceforth restrict attention to m ∈ Z_{≥0}.

Theorem 3.1. One has the discriminant formula

D(P_{m}) = ∏_{j=3,5,...}^{2m−1} j^{j(2m+1−j)^{2}/4}

and the resultant formulas

R(P_{m}, P_{m−1}) = ∏_{j=3,5,...}^{2m−3} j^{j(2m+1−j)(2m−1−j)/4}

R(P_{m}, P_{m−2}) = (2m − 1)^{Δ_{m−1}} ∏_{j=3,5,...}^{2m−5} j^{j(2m+1−j)(2m−3−j)/4}.

Here the products run over odd integers, as indicated.

Proof. Applying R(P_{m}, ·) to (3.1), (3.3), and (3.4) yields

(3.5) R(P_{m+1}, P_{m})R(P_{m}, P_{m−1}) = D(P_{m})^{2}
(3.6) R(P_{m+2}, P_{m})R(P_{m}, P_{m−2}) = ((2m − 1)(2m + 3))^{Δ_{m}}D(P_{m})^{2}
(3.7) R(P_{m+2}, P_{m})R(P_{m}, P_{m−1}) = (2m + 3)^{Δ_{m}}R(P_{m+1}, P_{m})D(P_{m}).

As abbreviations, define

d_{m} = D(P_{m})
r_{m} = R(P_{m}, P_{m+1})
(2m + 3)^{Δ_{m+2}}ρ_{m} = R(P_{m}, P_{m+2}),

so that

(2m − 1)^{Δ_{m}}ρ_{m−2} = R(P_{m−2}, P_{m}).

These abbreviations absorb most of the factors involving (2m + 3) and (2m − 1), and (3.5), (3.6), (3.7) can be written respectively as

(3.8)  d_{m}^{−2}r_{m} = r_{m−1}^{−1}
(3.9)  d_{m}^{−2}ρ_{m} = ρ_{m−2}^{−1}(2m + 3)^{−2m−3}
(3.10)  d_{m}^{−1}r_{m}^{−1}ρ_{m} = r_{m−1}^{−1}(2m + 3)^{−2m−3}.

The exponents of the three new variables in the three equations just displayed are as indicated in the following 3-by-3 matrix

d_{m} r_{m} ρ_{m}
(3.8) −2 1 0
(3.9) −2 0 1
(3.10) −1 −1 1

The inverse of this 3-by-3 matrix is

(3.8) (3.9) (3.10)
d_{m} −1  1  −1
r_{m} −1  2  −2
ρ_{m} −2  3  −2

Accordingly, to isolate the new variable d_{m} we form the multiplicative combination (3.8)^{−1}(3.9)^{1}(3.10)^{−1} and similarly for r_{m} and ρ_{m}. The result is

(3.11)  d_{m} = F_{m}^{1}r_{m−1}
(3.12)  r_{m} = F_{m}^{2}r_{m−1}
(3.13)  ρ_{m} = F_{m}^{3}r_{m−1}(2m + 3)^{−2m−3},

where we have defined F_{m} = r_{m−1}/ρ_{m−2}. Using (3.12) twice and (3.13) once we deduce a two-step recursion:

F_{m} = \frac{r_{m−1}}{ρ_{m−2}}
= \frac{F_{m−1}^{2}r_{m−2}(2m − 1)^{2m−1}}{F_{m−2}^{3}r_{m−3}}
= \frac{F_{m−1}^{2}F_{m−2}^{2}r_{m−3}(2m − 1)^{2m−1}}{F_{m−2}^{3}r_{m−3}}
= \frac{F_{m−1}^{2}(2m − 1)^{2m−1}}{F_{m−2}} .

By direct calculation one has F_{2} = 3^{3} and F_{3} = 3^{6}5^{5}. The unique solution of the two-step recursion satisfying these initial conditions is

F_{m} = ∏_{j=3,5,...}^{2n−1} j^{j(2m+1−j)}. (3.14)

Combining (3.14) with (3.11)-(3.13) and elementary steps gives the formulas in Theorem 3.1.

4. BiHermite polynomials

Our basic reference for biHermite polynomials is [NY], especially Theorem 4.2 which gives the quadratic relations we need. However we will renormalize the polynomials H_{m,n}^{NY} there via

H_{m,n}(x) = \frac{H_{m,n}^{NY}(x/\sqrt{3})}{\sqrt{3}^{mn+m(m−1)+n(n−1)} ∏_{j=1}^{m−1} j^{m−j} ∏_{j=1}^{n−1} j^{n−j}} .

The polynomials H_{m,n} are defined for m, n ∈ Z_{≥0} and characterized by

H_{m,0} = H_{0,n} = 1
H_{1,1} = x
(4.1)  nH_{m,n+1}H_{m+1,n−1} = H_{m,n}H′_{m+1,n} − H_{m+1,n}H′_{m,n}
(4.2)  mH_{m+1,n}H_{m−1,n+1} = H_{m,n}H′_{m,n+1} − H_{m,n+1}H′_{m,n}

One has the degree formula

degree(H_{m,n}) = mn

and the symmetry

(4.3)  H_{m,n}(ix) = i^{mn}H_{n,m}(x).

Finally Theorem 4.2 of [NY] gives not only the quadratic relations (4.1) and (4.2), but also

(4.4) H′_{m,n}H′_{m,n} − H_{m,n}H′′_{m,n} = −m(H_{m+1,n}H_{m−1,n} − H_{m,n}^{2})
(4.5) H′_{m,n}H′_{m,n} − H_{m,n}H′′_{m,n} = n(H_{m,n+1}H_{m,n−1} − H_{m,n}^{2}).

Define

e(j) = \{ \begin{matrix}j^{2} − 2(m − j)(n − j) & if j ≤ min(m, n) \\ min(m, n)^{2} & if min(m, n) ≤ j ≤ max(m, n) \\ (m + n − j)^{2} & if max(m, n) ≤ j. \end{matrix}

For n ∈ Z_{≥0} define also

n^{∗} = ∏_{j=1}^{n} j^{j}

n^{∗∗} = ∏_{m=1}^{n} m^{∗}.

We adopt the convention that (−1)^{∗} = 0^{∗} = (−1)^{∗∗} = 0^{∗∗} = 1.

Theorem 4.1. The discriminant of H_{m,n} is

D(H_{m,n}) = ∏_{j=2}^{m+n−1} j^{je(j)}.

Related resultants are

R(H_{m,n}, H_{m+1,n}) = \frac{(m + n − 1)^{∗∗}}{(m − 1)^{∗∗}m^{∗n}(n − 1)^{∗∗}} D(H_{m,n})
R(H_{m,n}, H_{m,n+1}) = \frac{(m + n − 1)^{∗∗}}{(m − 1)^{∗∗}n^{∗m}(n − 1)^{∗∗}} D(H_{m,n})
R(H_{m,n}, H_{m+1,n−1}) = \frac{(n − 1)^{∗m}}{m^{∗n}} D(H_{m,n}).

Proof. The theorem is true whenever mn = 0, as both sides of all four equations are 1. We totally order the remaining index set Z_{≥1}^{2} by saying

(μ, ν) < (m, n) ⇔ ( \begin{matrix}μ + ν & < & m + n \\ & or \\ μ + ν & = & m + n & and & μ & < & m \end{matrix} ) .

The proof proceeds inductively with respect to this order. So suppose the theorem is proved up through, but not including, the index (m, n) ∈ Z_{≥1}^{2}.

Abbreviate

a_{m,n} = R(H_{m,n}, H_{m+1,n})
b_{m,n} = R(H_{m,n}, H_{m,n+1})
c_{m,n} = R(H_{m,n}, H_{m+1,n−1})
d_{m,n} = D(H_{m,n}).

Abbreviate further

a = a_{m,n}  A = a_{m−1,n}
b = b_{m,n}  B = b_{m,n−1}
c = c_{m,n}  C = c_{m−1,n+1}
d = d_{m,n}.

The three index-pairs in the right column are all < (m, n). So we can regard A, B, C as old variables and a, b, c, d as new variables.

Applying R(H_{m,n}, ·) to (4.1), (4.2), (4.4), and (4.5), one gets

a^{−1}bcd^{−1} = n^{−mn}
ab^{−1}d^{−1} = C^{−1}m^{−mn}
a^{−1}d^{2} = Am^{mn}
b^{−1}d^{2} = Bn^{mn}.

We follow the basic pattern of (3.8)-(3.13). The exponents on the left side this time form a 4-by-4 matrix; this matrix is again invertible over the integers. Inverting this matrix, and taking corresponding multiplicative combinations, we get

a = A^{−3}B^{2}C^{2}n^{2mn}m^{−mn}
b = A^{−2}BC^{2}n^{mn}
c = A^{−2}B^{2}Cn^{mn}m^{−mn}
d = A^{−1}BCn^{mn}.

Plugging in the known formulas on the right and carrying out a number of elementary steps yields the formulas to be proved for the quantities on the left.

5. Okamoto polynomials

Our basic reference for Okamoto polynomials is again [NY], especially Theorem 4.2 which again give the quadratic relations we need. However we will reindex the polynomials Q_{m,n}^{NY} there by

Q_{m,n,r} = Q_{m,n} = Q_{m+n,n}^{NY}.

Here r is a redundant parameter always satisfying

m + n + r = 1.

We include r sometimes to see an S_{3} symmetry more clearly. For example,

d_{m,n} = \frac{1}{2}(m^{2} + n^{2} + r^{2} − 1) (5.1)

is the degree of Q_{m,n}.

The polynomials Q_{m,n} ∈ Z[x] are defined for m, n ∈ Z and characterized by the following equations:

Q_{0,0} = Q_{1,0} = Q_{0,1} = 1
Q_{1,1} = x
(5.2)  Q_{m+1,n}Q_{m−1,n} = (x^{2} + 2m + n − 1)Q_{m,n}^{2} +
Q′′_{m,n}Q_{m,n} − Q′_{m,n}Q′_{m,n}
(5.3)  Q_{m,n}(ix) = i^{d_{m,n}}Q_{n,m}(x).

One has the further symmetry

Q_{m,n,r} = Q_{r,m,n} = Q_{n,r,m}. (5.4)

This symmetry means that every Okamoto polynomial can be written Q_{m,n} with mn ≥ 0.

From Theorem 4.2 of [NY] one gets not only the quadratic relation (5.2), but also

(5.5) Q′_{m,n}Q_{m+1,n} = Q_{m,n}Q′_{m+1,n} − xQ_{m,n}Q_{m+1,n} + Q_{m+1,n−1}Q_{m,n+1}
(5.6)  0 = (3r − 1)Q_{m+1,n−1,r}Q_{m−1,n+1,r} +
(3m − 1)Q_{m,n+1,r−1}Q_{m,n−1,r+1} +
(3n − 1)Q_{m−1,n,r+1}Q_{m+1,n,r−1}.

For x, x′ ∈ Z with |x − x′| ≤ 1 define

f_{x,x′} = \{ \begin{matrix}∏_{j=2,5,...}^{3 min(x,x′)−4} j^{j(x−(j+1)/3)(x′−(j+1)/3)} & if x, x′ ≥ 2 \\ ∏_{j=4,7,...}^{3 min(|x|,|x′|)−2} j^{j(|x|−(j−1)/3)(|x′|−(j−1)/3)} & if x, x′ ≤ −2 \\ 1 & else. \end{matrix}

Clearly f_{x,x′} = f_{x,x′}. Also, straightforward calculation yields

(5.7)  f_{x−1,x}f_{x+1,x} = f_{x,x}^{2}
(5.8)  f_{x,x−1}^{2}f_{x+1,x+1} = f_{x,x+1}^{2}f_{x−1,x−1}(3x − 1)^{3x−1}.

Theorem 5.1. The discriminant of Q_{m,n,r} depends separately on m, n, and r via

D(Q_{m,n,r}) = f_{m,m}f_{n,n}f_{r,r}. (5.9)

If {m − m′, n − n′, r − r′} = {−1, 0, 1} then the corresponding resultant factors as

R(Q_{m,n,r}, Q_{m′,n′,r′}) = f_{m,m′}f_{n,n′}f_{r,r′}. (5.10)

Proof. A difference between the present Okamoto case and the biHermite case of §4 is that here we do not have a fact analogous to H_{m,0} = H_{0,n} = 1. Accordingly, we have to replace the induction there by a different inductive argument here.

Abbreviate

D_{m,n,r} = D(Q_{m,n,r})
R_{m,n,r}^{m′,n′,r′} = R(Q_{m,n,r}, Q_{m′,n′,r′}).

Applying R(Q_{m,n,r}, ·) to (5.2) and (5.5) respectively gives

(5.11)  D_{m,n,r}^{2} = R_{m,n,r}^{m+1,n,r−1}R_{m,n,r}^{m−1,n,r+1}
(5.12)  D_{m,n,r}R_{m,n,r}^{m+1,n,r−1} = R_{m,n,r}^{m+1,n−1,r}R_{m,n,r}^{m,n+1,r−1}.

In comparison with the biHermite case, we need to use a new relation to substitute for the lack of infinitely many initial conditions. We get this relation by applying R(Q_{m+1,n,r−1}, ·), R(Q_{m−1,n+1,r}, ·) and R(Q_{m,n−1,r+1}) in turn to (5.6), and taking the product of these three relations so that unwanted factors drop out. The result, after the substantial straightforward cancellation, is

R_{m,n−1,r+1}^{m−1,n,r+1}R_{m−1,n+1,r}^{m,n+1,r−1}R_{m+1,n,r−1}^{m+1,n−1,r} =

R_{m+1,n−1,r}^{m,n−1,r+1}R_{m−1,n,r+1}^{m−1,n+1,r}R_{m,n+1,r−1}^{m+1,n,r−1} · (5.13)

(3m − 1)^{3m−1}(3n − 1)^{3n−1}(3r − 1)^{3r−1}.

If one replaces the D’s and R’s in (5.11), (5.12), and (5.13) by f’s using the not-yet-known formulas (5.9) and (5.10) one gets relations among the f’s which need to be checked. For example, the f-version of (5.11) is

f_{m}^{m}f_{m}^{m}f_{n}^{n}f_{n}^{n}f_{r}^{r}f_{m}^{m} = f_{m+1}^{m}f_{n}^{n}f_{r}^{r−1}f_{m}^{m−1}f_{n}^{n}f_{r}^{r+1}.

This equation is true because of (5.7) applied twice, once with x = m and once with x = r. Similarly, the f-version of (5.12) is true because of (5.7)

Figure 5.1. Indices for the inductive step G_{3} → G_{4}. Vertices index discriminants and edges index resultants.

applied with x = n. Finally, the f-version of (5.13) is true because of three applications of (5.8), one each to x = m, n, and r.

It is best to present the inductive aspects of our argument geometrically. We will illustrate them by Figure 5.1. View m, n, and r as coordinate functions on the plane of Figure 5.1, satisfying m + n + r = 1. The central point in this figure is the non-lattice point (1/3, 1/3, 1/3). If (m, n, r) is a lattice point then we call the sum of the positive numbers among m, n, and r its height. So there are three points of height one, those with {m, n, r} = {1, 0, 0}. There are nine points of height two, three with {m, n, r} = {1, 1, −1} and six with {m, n, r} = {2, 0, −1}. In general, there are 6h − 3 lattice points of height h. Because of the S_{3} symmetry, we do not need to indicate which collections of parallel lines correspond to constant integral values of which variable.

Let G_{h} be the graph with vertices the lattice points of height ≤ h and edges connecting adjacent vertices. So Figure 5.1 shows all of G_{3} in gray, with edges dashed. It shows the rest of G_{4} in gray and black, edges not being dashed. The union of the G_{h} is an infinite planar graph we call G.

Figure 5.2. Diagrams indicating factors present in Equations (5.11), (5.12), (5.13), respectively.

To the vertex (m, n, r) ∈ G there corresponds the real number D_{m,n,r}. To the edge in G connecting (m, n, r) with the adjacent vertex (m′, n′, r′) corresponds the real number R_{m,n,r}^{m′,n′,r′}. Note that the two functions

D : Vertices → R_{≥0}
R : Edges → R_{≥0}

are S_{3}-invariant. So are the formulas on the right side of (5.9) and (5.10). The diagrams in Figure 5.2 correspond to relations (5.11), (5.12), and (5.13) respectively. Consider the edge graph rigidly embedded somehow into G; “rigidly” means here “no bending in the middle”; however different orientations are allowed, according to the S_{3} symmetry. One has a discriminant corresponding to the middle vertex and two resultants corresponding to the two edges. If our formula is true for two of these quantities then it is true for the third. Similarly, consider the corner graph rigidly embedded into G. Here one has a discriminant and three resultants. If our formula is true for three of these quantities it is true for the fourth. Finally, consider the hexagon graph rigidly embedded. Here one has six resultants corresponding to the six edges. If our formula is true for five of these six resultants it is true for the last.

We begin the induction at G_{3}, for which the truth of the formulas can be checked by direct computation. Next we use the edge relation to extend the domain on which the formula is known to the edges which have a vertex in G_{3}; these 36 edges are drawn in gray. Next we use the hexagon relation to extend this checked domain to the remaining edges except the twelve remaining edges which are incident upon one of the six corners; these 9 edges are drawn as thin black lines. Third, all vertices except the six corners are obtained by using corner relations. Fourth, the twelve remaining edges are obtained by edge relations. Fifth, the six vertices are obtained by corner relations. These last three parts of G_{4} are all shown in black, with thick edges. The remaining inductive steps G_{n} → G_{n+1} are done in the same way.

6. Larger picture

It is known that Painlevé I has no rational solutions. It is also known that the rational functions discussed in §1 are the only rational solutions to Painlevé II and Painlevé IV. Finally, there are complete lists of rational solutions to Painlevé III, V, VI, coming from polynomials analogous to P_{m}, H_{m,n}, and Q_{m,n}. The extreme case of Painlevé VI is particularly interesting; see [NOOU] for explicit polynomials and [Ma] for the completeness result.

The Painlevé polynomials relating to Painlevé III, V, and VI fit into continuous families. They can be divided into two types:

• Deformations of Yablonsky-Voboriev polynomials. We call these triangular Painlevé polynomials.

• Deformations of biHermite polynomials. We call these biclassical polynomials because the case n = 1 gives the classical Hermite-Laguerre-Jacobi hierarchy.

We have computationally investigated all these Painlevé polynomials in considerable detail. For example, from our computations we have extracted conjectural discriminant formulas for the remaining Painlevé polynomials. Presumably these discriminant formulas have proofs similar to the three presented here. The discriminant formulas in the biclassical case reduce for n = 1 to classical formulas dating back to Hilbert [Hi] and Stieltjes [St].

We are quite convinced that all these discriminant formulas form just part of a larger picture. Here are very brief sketches of topics we plan to treat more fully in the future.

Complex roots of P_{m}, H_{m,n}, and Q_{m,n}. In Figure 6.1, each of the four squares represents the region |Re(x)|, |Im(x)| ≤ 5 in the complex x-plane. In each square, we have plotted the roots of the indicated Painlevé polynomial. In general, one has the following experimental observations. The roots of P_{m} approximately form a triangle with m roots on a side. The roots of H_{m,n} approximately form an m × n rectangle for m/n ∼ 1; if m/n is far from 1 then the rectangle becomes quite distorted but the rectangular structure remains clearly visible. For m, n ≥ 0, the roots of Q_{m,n} and Q_{−n,−m} approximately form an m × n rectangle appropriately framed by triangles in accordance with the degree formulas

(6.1)  degree(Q_{m,n}) = mn + 2Δ_{m−1} + 2Δ_{n−1}
(6.2)  degree(Q_{−n,−m}) = mn + 2Δ_{m} + 2Δ_{n}.

Figure 6.1. Complex roots of four Painlevé polynomials

Here both (6.1) and (6.2) are rewritten versions of the symmetric degree formula (5.1). In all three cases the roots behave so regularly that, after separating out polynomials according to their indices modulo 3, 2, and 2, respectively, one can interpolate roots with m and n ∈ R.

p-adic roots of P_{m}, H_{m,n}, and Q_{m,n}. Our discriminant formulas say that 2 doesn’t divide D(P_{m}) and 3 doesn’t divide D(Q_{m,n}). The separable polynomials P_{m} ∈ F_{2}[x] and Q_{m,n} ∈ F_{3}[x] factor in an unusually regular way.

In all other cases, one can use Newton polygons to investigate the ramification at p. These polygons behave very regularly. Combining Newton polygon information with our discriminant formulas gives prime-by-prime bounds on the algebra discriminants of Q[x]/P_{m}(x), Q[x]/H_{m,n}(x), and Q[x]/Q_{m,n}(x). The simplest behavior is for large primes dividing the polynomial discriminant, namely the primes which contribute to only one factor in the polynomial discriminant formula. The p-adic roots exhibit sufficient periodicity that it makes sense to interpolate them with m and n in Z_{p}.

Complex monodromy of Painlevé III, V, and VI polynomials. As mentioned above, these remaining Painlevé polynomials depend on free parameters. Geometrically, the new parameters let complex roots as in the top two frames of Figure 6.1 move. Our conjectural discriminant formulas identify the locus D in the complex parameter space P where complex roots come together. Computation suggests extremely regular behavior for how roots are interchanged as one moves the complex parameters through P − D.

References

[Co] Robert Conte, editor, The Painlevé Property: One Century Later, Springer-Verlag (1999).

[Hi] David Hilbert, Über die Diskriminante der im Endlichen abbrechenden hypergeometrischen Reihe, Journ. f. reine u. angew. Mathematik 103, (1888) 337-345.

[IKSY] Latsunori Iwasaki, Hironobu Kimura, Shun Shimomura, and Masaaki Yoshida, From Gauss to Painlevé: A Modern Theory of Special Functions, Vieweg (1991).

[KYO] Kenji Kajiwara, Kazushi Yamamoto, and Yasuhiro Ohta, Rational solutions for the discrete Painlevé II equation, Physics Letters A 232 (1997) 189-199.

[Ma] Marta Mazzocco, Rational solutions of the Painlevé VI equation, J. Phys. A: Math. Gen. 34, (2001) 2281-2294.

[Mu] Yoshihiro Murata, Rational solutions of the second and fourth Painlevé equations, Funkcial. Ekvac. 28 (1985), no. 1, 1-32.

[NOOU] Masatoshi Noumi, Soichi Okada, Kazuo Okamoto, and Hiroshi Umemura, Special polynomials associated with the Painlevé equations II, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), 349-372, World Sci. Publishing.

[NY] Masatoshi Noumi and Yasuhiko Yamada, Symmetries in the fourth Painlevé equation and Okamoto polynomials, Nagoya Math J. 153 (1999), 53-86.

[St] Thomas Stieltjes, Sur les polynômes de Jacobi, C.-R. Acad. Sci., 100 (1885), 620-622.

[Um] Hiroshi Umemura, The Painlevé equation and classical functions, Sugaku Expositions 11 (1998), no 1, 77-100.

[Vo] A. P. Vorobiev, On the rational solutions of the second Painlevé equation, Differential Equations I (1965), 58-59.

[Ya] A. I. Yablonsky, Vestsi Akad. Navuk BSSR Ser. Fiz.-Tekh. Navuk No. 3 (1959), 30-35.

Division of Science and Mathematics, University of Minnesota-Morris, Morris, Minnesota, 56267

E-mail address: roberts@morris.umn.edu
URL: http://cda.morris.umn.edu/~roberts/